What SI unit describes the mass of an atom or molecule?
Kilograms or grams.
The official SI unit of mass is the kilogram. Due to the small amounts involved at a molecular level we may use grams (0.001 kg) for things like molecular weights.
Actual masses of individual atoms or molecules will be very small, requiring scientific notation to avoid writing a lot of leading zeros. Thus it doesn't really matter whether you use grams or kilograms, as the required exponent will just be a number that is 3 more (and negative) for kilograms compared to grams.
The "dalton" may also be used, but strictly speaking it is not an SI unit.